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Real versus Complex Volumes on Real Algebraic Surfaces
Arnaud Moncet
To cite this version:
Arnaud Moncet. Real versus Complex Volumes on Real Algebraic Surfaces. International Mathematics Research Notices, Oxford University Press (OUP), 2012, 16, pp.3723-3762. �10.1093/imrn/rnr143�.
�hal-00599628v2�
ON REAL ALGEBRAIC SURFACES
ARNAUD MONCET
Abstract. LetX be a real algebraic surface. The comparison between the volume ofD(R)andD(C)for ample divisorsDbrings us to define the con- cordance α(X), which is a number between 0and 1. This number equals 1 when the Picard numberρ(XR)is1, and for some surfaces with a “quite sim- ple” nef cone, e.g. Del Pezzo surfaces. For abelian surfaces,α(X)is1/2or1, depending on the existence or not of positive entropy automorphisms onX. In the general case, the existence of such an automorphism gives an upper bound forα(X), namely the ratio of entropies htop(f|X(R))/htop(f|X(C)). More- overα(X)is equal to this ratio when the Picard number is2. An interesting consequence of the inequality is the nondensity ofAut(XR)inDiff(X(R))as soon asα(X)>0. Finally we show, thanks to this upper bound, that there exist K3 surfaces with arbitrary small concordance, considering a deformation of a singular surface of tridegree(2,2,2)inP1×P1×P1.
1. Introduction
LetX be a real projective variety with a fixed Riemannian metric. The goal of this paper is to compare volumes of real subvarietiesY(R)and of their complexifi- cationsY(C). As will be seen, this is closely related to the question of comparing real and complex dynamics of automorphisms of XR.
1.1. Projective space. Consider the projective space X = PdR, equipped with the Fubini–Study metric. Let Y be a real subvariety of PdR of dimension k. By Wirtinger’s formula (see [15, p. 31]), the volume ofY(C)satisfies
volC(Y) = deg(Y) volC(Pk). (1) For the volume ofY(R), the Cauchy–Crofton formula enables us to show that
volR(Y)≤deg(Y) volR(Pk) (2) and to characterize the case of equality (see Appendix A). This gives the following proposition.
Proposition 1.1. Let Y be a real k-dimensional algebraic subvariety of the pro- jective spacePdR. With respect to the Fubini–Study metric, we have
volR(Y)
volR(Pk)≤ volC(Y)
volC(Pk). (3)
Furthermore, equality is achieved if and only if Y is the union of deg(Y)real pro- jective subspaces.
Date: July 2011.
1
As a consequence, if Vk(δ) denotes the set of real subvarieties ofPdR of dimen- sionkand degreeδ, then for anyY0∈ Vk(δ)we have
Ymax∈Vk(δ)volR(Y) =δvolR(Pk) = volR(Pk)
volC(Pk)volC(Y0). (4) 1.2. General case. Now X is an arbitrary d-dimensional real algebraic variety.
We assume that it isprojective, smooth, irreducible and that the real locus X(R) isnot empty. LetY be ak-dimensional real algebraic subvariety ofX. Denote the volume ofY(R)byvolR(Y), and that ofY(C)byvolC(Y), both with respect to a fixed Riemannian metric onX(C).
Notation. LetV(Y)be the family of real algebraic subvarietiesZ such thatY(C) andZ(C)have the same homology class inH2k(X(C);Z). Then forK =RorC we set
mvolK(Y) = max
Z∈V(Y)volK(Z). (5)
When the Riemannian metric comes from the Fubini–Study metric on some projective spacePn in whichX is embedded, we get Inequality (3). Since two Rie- mannian metrics are comparable (by compactness ofX(C)), we obtain the following proposition.
Proposition 1.2. Let X be a real algebraic variety, equipped with an arbitrary Riemannian metric. For any k∈N∗ there exists a constantCk >0 depending on the choice of the metric, such that
mvolR(Y)≤CkmvolC(Y) (6)
for allk-dimensional subvarietiesY ofX.
Now we would like to know for which nonnegative exponents α we can write inequalities such as mvolR(Y)≥ CkmvolC(Y)α, with Ck independent of Y. We restrict ourselves to codimension 1 subvarieties, that is, effective divisors. The notion of homology class in H2d−2(X(C);Z)is dual to that of (first) Chern class inH2(X(C);Z), which is preferred in what follows.
Definition 1.3. LetA(X)be the set of nonnegative exponentsαfor which there existC >0and q∈N∗ such that
mvolR(D)≥CmvolC(D)α (7)
for all real ample divisorsDwhose Chern classes areq-divisible. The upper bound ofA(X)is theconcordance ofX, and is denoted byα(X). We say the concordance isachieved whenα(X)is contained inA(X).
The setA(X), and thus the concordanceα(X), only depend onX, and not on the choice of a particular metric. All metrics will be Kähler metrics, so that the number volC(D) only depends on the Chern class of D; thus we write volC(D) instead ofmvolC(D).
As will be seen in Section 2.3, the concordance α(X)can only take values be- tween 0 and1, and the setA(X) is an interval of the form[0, α(X)]or [0, α(X)), whether the concordance is achieved or not.
1.3. Examples. Equation (4) implies that the concordance of the projective space is 1. More generally we prove in Section 2.4 thatα(X) = 1 as soon as the closed convex coneNef(XR)of real nefR-divisors is generated by finitely many divisorsDj
withmvolR(Dj)>0. This is the case when the real Picard numberρ(XR)is1. As a special case, the concordance of a curve is always1. Thus, nontrivial cases (those with α(X)<1) can only occur when both the dimension and the Picard number are at least2.
In this paper, we focus on the case of surfaces, which already include many interesting examples. Among them, tori are the simplest surfaces for which the concordance is not always1 (cf §4):
Theorem 1.4. Let X be a real abelian surface. The real Picard numberρ(XR)is equal to 1,2 or3, and we have the following values for concordance:
(1) If ρ(XR) = 1, thenα(X) = 1.
(2) If ρ(XR) = 2, thenα(X) = 1 or1/2, depending on the existence or not of real elliptic fibrations on X.
(3) If ρ(XR) = 3, thenα(X) = 1/2.
In all cases the concordance is achieved.
In Section 5.3 we show that there exist surfaces with arbitrary small concordance.
More precisely, we prove the following result.
Theorem 1.5. There is a family(Xt)t∈(0,1] of real K3 surfaces embedded in(P1)3 such that
t→0limα(Xt) = 0. (8)
1.4. Dynamics of automorphisms. LetX be a real algebraic surface. We de- note byAut(XR)the group of (real) automorphisms onX, that is, biholomorphic maps f : X(C) → X(C) that commute with the antiholomorphic involution σ.
ForK=RorC the induced self-map onX(K)is denoted byfK.
The dynamics of automorphisms on complex surfaces has been broadly studied in the last decades (one may refer to the references given in the surveys [10] and [3]).
Let us remember a few facts:
(1) The entropy htop(fC) is entirely expressed in terms of the action on the cohomology, according to [17] and [34]; namely, it is equal to the logarithm of the spectral radius (called thespectral logradius in what follows) of the induced mapf∗ onH2(X(C);R).
(2) Automorphisms that have positive entropy, also called hyperbolic type au- tomorphisms, can only occur on tori, K3 surfaces, Enriques surfaces, and (nonminimal) rational surfaces, or on blow-ups of such surfaces at periodic orbits [8]. Moreover, examples are known on each of these types of surfaces.
(3) For hyperbolic type automorphisms we have htop(fC) ≥ log(λ10) [25], λ10≃1,17628081 being the Lehmer number. Moreover, this bound is achieved on some rational surfaces [4] [25] and on some K3 surfaces (C.
McMullen gives a nonprojective example in [26], and announces that there also exists a projective example).
On the other hand the dynamics onX(R)is not as well understood, for we do not dispose of equivalent tools to study it. For instance the entropyhtop(fR)cannot be deduced from the action on cohomology; it is bounded from below by the spectral radius offR∗ onH1(X(R);R)[22], and from above byhtop(fC), but may vary within
this interval. In particular, we see that, for hyperbolic type automorphisms, the ratiohtop(fR)/htop(fC)is a number between0and1(for tori it always equals1/2:
cf Proposition 4.4). As proved by Bedford and Kim in [5], this ratio may happen to be equal to1 for some rational1surfaces.
Question 1.6. Is there an example of a real hyperbolic type automorphism on a K3 or Enriques surfaceX for whichhtop(fR) = htop(fC)?
Question 1.7. Is there an example of a real hyperbolic type automorphism on a surfaceX for whichhtop(fR) = 0?
Remark 1.8.In [5] the authors prove the maximality of entropy using only homology of real algebraic curves. In this text, I rather use their volumes, which provide a finer measure than their homology classes.
In Section 3 we use a theorem due to Yomdin [34] in order to highlight a link be- tween concordance and this ratio of entropies (which is used to prove Theorems 1.4 and 1.5).
Theorem 1.9. Let X be a real algebraic surface. Assume that there exists a real hyperbolic automorphismf onX. Then
α(X)≤htop(fR)
htop(fC). (9)
Moreover, this inequality becomes an equality whenρ(XR) = 2.
Corollary 1.10. Let f be a real automorphism of a real algebraic surface X. If htop(fR)>0, then
htop(fR)≥λ10α(X), (10)
where λ10 denotes the Lehmer number, that is, the largest root of the polynomial x10+x9−x7−x6−x5−x4−x3+x+ 1.
Whenα(X) >0, these results enables us to show nondensity and discreteness results for Aut(XR)in the group of diffeomorphisms of X(R), as well as in some of its subgroups (§6).
Acknowledgements. The work presented in this paper is part of my PhD thesis.
I am especially grateful to my thesis advisor Serge Cantat who suggested the topic, encouraged my progress, and patiently read and corrected this text many times during its preparation. I also thank the referee for giving this paper an exceptionally prompt and thorough reading, and for his helpful and positive comments. I would also like to thank Sébastien Gouëzel, Yutaka Ishii, Frédéric Mangolte and Anton Zorich for many useful discussions. Finally I would like to thank Laura DeMarco for having invited me to the Workshop on Dynamics at the University of Illinois at Chicago in may 2010, as well as all the organizers of this conference.
2. First Properties of Concordance
2.1. Conventions and notations. In what follows,X denotes the ambient real algebraic variety, and d its dimension. Moreover, X is always supposed to be projective, smooth, irreducible and with nonempty real locus. The set X(R) is then a real analytic d-dimensional manifold, with a finite number of connected
1Throughout the text, rational means rational overC.
components. In contrast, we make no particular assumption for subvarietiesY ofX. The antiholomorphic involution that defines the real structure on X is denoted byσX, or simply byσwhen no confusion is possible.
The cohomology groups Hk(X(C);Z) are implicitly taken modulo torsion, so that we can consider them as lattices inHk(X(C);R).
Thecomplex Néron–Severi groupof X, denoted by N1(XC;Z), is the subgroup ofH2(X(C);Z)whose elements are Chern classes of divisors onX(C). We denote by[D] the (first) Chern class of a divisorD. By the Lefschetz theorem on (1,1)- classes (see [15, p. 163]), we have
N1(XC;Z) = H1,1(X(C);R)∩H2(X(C);Z). (11) The real Néron–Severi group of X, denoted by N1(XR;Z), is the subgroup ofN1(XC;Z)whose elements are classes of real divisors. Recall that
[σ(D)] =−σ∗[D] (12)
for any complex divisor D (see [31, §I.4]), where σ∗ denotes the involution on H2(X(C);Z)induced by the complex conjugationσ. Hence
N1(XR;Z) ={θ∈N1(XC;Z)|σ∗θ=−θ}, (13) BothN1(XC;Z)andN1(XR;Z)are free abelian groups of finite rank. Their re- spective ranks are thecomplexandreal Picard numbersofX, denoted byρ(XC)and ρ(XR). ForK=RorC we denote byN1(XK;R)the subspace of H1,1(X(C);R) spanned byN1(XK;Z); it has dimension ρ(XK).
WhenXis a surface, the intersection form gives rise to a nondegenerate quadratic form onH2(X(C);R), with integral values onH2(X(C);Z). By the Hodge index theorem, its signature on the subspaceN1(XK;R)is(1, ρ(XK)−1). Consequently, the positive cone for the intersection form has two connected components, one of which contains classes of ample divisors. This component is an open convex cone inN1(XK;R), denoted byPos(XK). Other convex cones inN1(XK;R)have their own interest and are used throughout this text, like the ample coneAmp(XK), the nef coneNef(XK), which is its closure, and the cone of curvesNE(XK)(for surfaces it is the same as the pseudo-effective cone), which is the dual of the last one. For all these notions, we refer to [20].
2.2. Positivity of volumes. From now on we fix a Kähler metric on the complex manifoldX(C). Its Kähler form is denoted by κ.
2.2.1. Complex volumes. LetD be an effective divisor onX. The volume ofD(C) only depends on the Chern class[D]. More precisely,
volC(D) = 1
(d−1)![κd−1]·[D]>0. (14) Proposition 2.1. There exists a positive constantK such that
volC(D)≥K (15)
for all effective divisors D6= 0.
Proof. As all Riemannian metrics are equivalent, it is enough to show the inequality when the metric is the Fubini–Study metric on Pn ⊃X. In this case the volume ofD(C)is proportional to the degree ofDas a subvariety ofPn, which is a positive integer. Thus we get the lower bound withK= volC(Pd−1)>0.
2.2.2. Real volumes. LetD be a real effective divisor onX. Although the volume ofD(C)is always positive, it may happen that volR(D) = 0 for some divisorsD.
For instance, on X = PdR, for any even degree δ the divisor Dδ given by the equation Pd
j=0Zjδ = 0 has an empty real locus, hence volR(Dδ) = 0. Yet this divisor is numerically (and even linearly) equivalent toDδ′ given byPd
j=1Zjδ=Z0δ, and we havevolR(D′δ)>0.
Thus what is important is not the positivity ofvolR(D), but that ofmvolR(D).
Remember thatmvolR(D) = max{volR(D′)|D′ ∈ V(D)}, whereV(D)is the set of real effective divisors (numerically) equivalent toD. Since
V(D1+D2)⊃ V(D1) +V(D2), (16) the function mvolR is superadditive on the set of real effective divisors. In partic- ular, for allk∈N∗,
mvolR(kD)≥kmvolR(D). (17)
Proposition 2.2. Let D be a real effective divisor such that the linear system|D| contains a pencil, that is, h0(X,OX(D))≥2. ThenmvolR(D)>0.
Proof. Let(Dλ)λ∈P1(C) be a real pencil in|D| (in this context, real meansDλ = σ(Dλ)). By Bertini’s theorem [15, p.137] there is a finite set S ⊂ P1(C) such that, for all λ /∈ S, the subvarietyDλ(C)is smooth away from the base locus B of the pencil (Dλ)λ∈P1(C). Let P ∈ X(R)\ B∪S
λ∈SDλ
. Then there exists λ inP1(R)\S such that the pointP is on (the support of) the divisorDλ. AsDλ is smooth atP, the real locusDλ(R)contains an arc aroundP, and thusmvolR(D)≥
volR(Dλ)>0.
This proposition applies, for instance, whenDis very ample. In contrast, it may happen thatmvolR(D) = 0for some effective divisors that are not ample, as shown in the two following examples. It is for this reason that we restrict ourselves to ample divisors in the definition of concordance.
Example 2.3. Let X be the variety obtained by blowing up PdR at two (distinct) complex conjugate points, and let E be the exceptional fiber of the blow-up. For any k ∈ N∗, we have V(kE) = {kE}, thus mvolR(kE) = 0, forE(R) is empty.
Nevertheless, observe that[E] is not in the closure of the cone Pos(XR), since its self-intersection is negative.
Example 2.4. LetC be a real smooth quartic inP2R such thatC(R)is empty (for instance, the one given byZ04+Z14+Z24= 0). Take8pairs of complex conjugate points(Pi, Pi)onC in such a way that the linear class ofP
i(Pi+Pi)− OP2(4)|C
is not a torsion point ofPic0(C). Letπ:X→P2 be the blow-up morphism above these16points (defined overR) and letC′ be the strict transform ofCin X. For all divisors D in V(kC′) the curveπ∗D has degree 4k and passes through the16 blown-up points with multiplicity at least k. Then the choice of the points Pi
implies that π∗D = kC, hence D = kC′. So we see that mvolR(kC′) = 0 for allk∈N∗. HereC′ is a nef divisor that is not ample as an irreducible divisor with self-intersection0 (see [20, §1.4]).
2.3. The interval A(X). Remember that (Definition 1.3)A(X) is the set of ex- ponentsα≥0for which there existC >0 andq∈N∗ such that, for all real ample divisorsDwith[D]q-divisible, we havemvolR(D)≥CvolC(D)α. This set depends only onX, and not on the choice of the metric.
Lemma 2.5. Assume that α∈ A(X). Then β∈ A(X)for all0≤β < α.
Proof. By Proposition 2.1, there existsK >0such thatmvolC(D)≥Kfor all real effective divisorsD. When[D]is q-divisible, we then have
mvolR(D)≥CvolC(D)α≥CKα−βvolC(D)β, (18)
and soβ is inA(X)too.
As a consequence, A(X) is an interval of the form [0, α(X)] or [0, α(X)). By definition,α(X)is the concordance of X.
Lemma 2.6. Let X be a real algebraic variety (with X(R) 6= ∅). The concor- danceα(X)is in the interval[0,1].
Proof. Let α ∈ A(X). By Proposition 1.2, there exists a positive C′ > 0 such that mvolR(D) ≤ C′volC(D) for all real ample divisors D. When [D] is also q-divisible, we obtain, for allk∈N∗,
CvolC(kD)α≤mvolR(kD)≤C′volC(kD). (19) Ifα >1, this contradictslimk→+∞volC(kD) = +∞. 2.4. Examples of varieties with concordance 1. We have seen in the intro- duction that α(PdR) = 1. More generally, the concordance is1 when the structure of the nef cone is “simple”.
Proposition 2.7. LetX be a real algebraic variety. Assume that the coneNef(XR) is polyhedral, with extremal rays spanned by classes [Dj] such thatmvolR(Dj)>0.
Then the concordanceα(X)is1, and it is achieved.
Proof. DefineC= minj(mvolR(Dj)/volC(Dj))>0. The classes[Dj]span a finite index subgroup ofN1(XR;Z). Denote byqthis index. SinceAmp(XR)⊂Nef(XR), every real ample divisor D with [D] q-divisible is equivalent to a divisor of the formP
jkjDj, where thekj’s are nonnegative integers. Hence mvolR(D)≥X
j
kjmvolR(Dj)≥CX
j
kjvolC(Dj) =CvolC(D). (20)
We can then conclude that1is contained inA(X).
Corollary 2.8. All real algebraic varieties X withρ(XR) = 1have concordance1, and this one is achieved.
Corollary 2.9. Let X be a real Del Pezzo surface. The concordance of X is 1, and it is achieved.
Proof of Corollary 2.9. By definition, a surface is Del Pezzo when its anticanonical divisor −KX is ample. The cone of curves NE(XR) is then rational polyhedral by the cone theorem (see [20, 1.5.33, 1.5.34]). Thus its dual coneNef(XR) is also rational polyhedral.
Lemma 2.10. Let D be a nef divisor on a Del Pezzo surface X, which is not numerically trivial. Then the linear system|D| contains a pencil.
Proof. The proof is a simple application of the Riemann–Roch formula:
h0(X,OX(D))−h1(X,OX(D))+h2(X,OX(D)) =χ(OX)+1
2(−KX·D+D2). (21) As−KX is ample andDis nef,−KX·D >0 andD2≥0. By Serre duality, we geth2(X,OX(D)) =h0(X,OX(KX−D)) = 0, because D·(KX−D)<0withD nef. We conclude thath0(X,OX(D))> χ(OX) = 1 (the last equality follows from
the rationality ofX).
Consequently, we see, by Proposition 2.2, that the extremal rays ofNef(XR)are spanned by classes[Dj]withmvolR(Dj)>0, and thus we can apply Proposition 2.7
to get the desired result.
3. Concordance and Entropy of Automorphisms
From now onX is a real algebraic surface equipped with a Kähler metric whose Kähler form is denoted byκ.
For any differentiable dynamical systemg:M →M on a compact Riemannian manifold, let htop(g) denote the topological entropy, and χtop(g) the topological Liapunov exponent, that is,
χtop(g) = lim
n→+∞
1
nlogkDgnk∞, (22) where the notationkDgk∞stands formaxx∈MkDg(x)k, the norm being taken with respect to the Riemannian metric. Note that the numberχtop(g)does not depend on the choice of the Riemannian metric.
Whenf ∈Aut(XR)is a real automorphism ofX, we are going to look at both differentiable dynamical systemsfC:X(C)→X(C)andfR:X(R)→X(R).
We denote by f∗ the inverse of the map f∗ induced by f on H2(X(C);Z), so that the operation f →f∗ is covariant. The linear mapf∗ is an isometry for the intersection form and preserves the Hodge structure: we say it is aHodge isometry.
Furthermore it is also compatible with the direct image of divisorsD, which means thatf∗[D] = [f∗D]. Hencef∗preserves the subgroupsN1(XK;Z)forK=RorC.
We still denote byf∗ the restriction off∗ to all the subgroups or subspaces (when extended byRorC) that are preserved.
The spectral radius off∗(a priori onH2(X(C);R)) is denoted byλ(f). By the theorem of Gromov and Yomdin recalled in the introduction, we have
htop(fC) = log(λ(f)). (23)
This spectral radius is actually achieved on the subspaceN1(XR;R)(cf Remark 3.2).
3.1. Complex volume of the iterates of a divisor.
Theorem 3.1. Let f be an automorphism of a complex algebraic surfaceX. For all ample divisors D we have
n→+∞lim 1
nlog (volC(f∗nD)) = htop(fC) = log(λ(f)). (24) Proof. Wirtinger’s equality gives (cf (14))
volC(f∗nD) =f∗n[D]·[κ]. (25)
If λ(f) = 1, the sequence (kf∗n[D]k)n∈N has at most a polynomial growth (ac- tually it is at most quadratic [14]), as well as(volC(f∗nD))n∈N. Hence
n→+∞lim 1
nlog (volC(f∗nD)) = 0 = log(λ(f)). (26) Ifλ(f)>1, since[D]is in the ample cone, the sequence fn
∗[D]
λ(f)n
n∈N converges to the classθ of a positive closed current, by [9]. In particular
n→+∞lim
volC(f∗nD)
λ(f)n =θ·[κ]>0, (27) and thenlimn→+∞ 1
nlog (volC(f∗nD)) = log(λ(f)).
Remark 3.2. If, moreover, the surfaceX, the automorphismf, and the divisorD are defined overR, then the classθis inN1(XR;R)(as a limit of classes that are in this closed subspace), and satisfiesf∗θ=λ(f)θ. Thusλ(f)is an eigenvalue off∗
restricted toN1(XR;R).
Remark 3.3. For varieties that have arbitrary dimensiond, the formula
n→+∞lim 1
nlog (volC(f∗nD)) = log(λ(f)) (28) still holds. But this is not necessarily equal to the entropy, which is the spec- tral logradius2on the whole cohomology,a priori distinct from the spectral logra- diuslog(λ(f))onH2(X(C);R).
3.2. An upper bound for real volume of the iterates of a divisor.
Theorem 3.4. Letf be a real automorphism of a real algebraic surfaceX. For all ample real divisors D we have
lim sup
n→+∞
1
nlog (mvolR(f∗nD))≤htop(fR). (29) The proof of this result relies on [34, Theorem 1.4], which gives a lower bound for entropy in terms of volume growth. It is here stated in the particular case of dimension1submanifolds.
Theorem 3.5 (Yomdin). Let M be a compact Riemannian manifold, g:M →M be a differentiable map and γ: [0,1]→M be an arc, each of classCr, withr≥1.
Then
lim sup
n→+∞
1
nlog (length(gn◦γ))≤htop(g) +2
rχtop(g). (30) In particular when the regularity isC∞, then
lim sup
n→+∞
1
nlog (length(gn◦γ))≤htop(g). (31) Looking carefully at the proof in Yomdin [34], one sees that this result can be improved to the case when we consider a family ofCr-arcs(γj)j whose derivatives are uniformly bounded to the orderr, that is, there is a positive number K such thatkγj(k)(t)k ≤K for allj,t∈[0,1]andk≤r. Under these assumptions we have
lim sup
n→+∞
1 nlog
maxj {length(gn◦γj)}
≤htop(g) +2
rχtop(g). (32)
2Recall thatspectral logradius stands for the logarithm of the spectral radius.
We also use the following lemma, which can be found in Gromov [16, 3.3].
Lemma 3.6(Gromov). LetY be the intersection of an algebraic affine curve inRd with [−1,1]d. For any r ∈ N∗ there exist at most m0 Cr-arcs γj : [0,1] → Y, wherem0 is an integer depending only on d,r, anddeg(Y), such that
(1) Y =S
jγj([0,1]);
(2) kγj(k)(t)k ≤1 for allj,t∈[0,1]andk≤r;
(3) allγj’s are analytic diffeomorphisms from (0,1)to their images;
(4) the images of the γj’s can only meet on their boundaries.
Proof of Theorem 3.4. Inequality (29) does not depend on the choice of a particular metric onX, so we can consider an embeddingX⊂PdRand take the metric induced by Fubini–Study on X. The projective space Pd(R) is covered by the (d+ 1) cubes Qk, k ∈ {0,· · ·, d} given in homogeneous coordinates by |Zk|= maxj|Zj|. Each of theseQk is located in the affine chart Uk ={Zk 6= 0} ≃Rd, and in this chart it is identified with[−1,1]d.
The degree of D as a subvariety of Pd only depends on the Chern class [D].
Therefore we can apply Lemma 3.6 to any divisorD′∈ V(D), intersected with one of the Qk’s: any real locus of D′ ∈ V(D)is covered by at mostm1 Cr-arcsγD′,j, the integer m1= (d+ 1)m0 being independent of D′, such thatkγ(k)D′,jk∞≤K for allk≤r, whereris a fixed positive integer andKa positive constant (which comes from the comparison of Euclidean and Fubini–Study metrics on[−1,1]d). Now we apply (32) to obtain
lim sup
n→+∞
1
nlog (mvolR(f∗nD))≤lim sup
n→+∞
1 nlog
m1max
D′,j{length(fRn◦γD′,j)}
≤htop(fR) +2
rχtop(fR).
(33)
Since the regularity of bothX(R)andfR isC∞, we may take the limit asrgoes
to+∞and get the desired inequality.
Remark 3.7. Yomdin’s theorem (as well as its version in family) and Gromov’s lemma still hold for arbitrary dimensional submanifolds. Therefore the proof of Theorem 3.4 can be adapted whenX is a variety with higher dimension.
3.3. An upper bound for concordance.
Theorem 3.8. LetX be a real algebraic surface and letf be a real hyperbolic type3 automorphism ofX. Then
α(X)≤htop(fR)
htop(fC). (34)
Proof. Letαbe an exponent in the intervalA(X). This means that there areq∈N∗ and C > 0 such that mvolR(D) ≥ CvolC(D)α for all real ample divisors D with [D] q-divisible. For such a divisor, f∗n[D] is also q-divisible for all n ∈ N,
3Recall thathyperbolic typejust means thathtop(fC)>0.
and by Theorems 3.1 and 3.4 we get htop(fR)≥lim sup
n→+∞
1
nlog mvolR(f∗nD)
≥lim sup
n→+∞
1
n(logC+αlog volC(f∗nD))
=αhtop(fC).
(35)
Then we take the limit asα→α(X)and we obtain (34).
3.4. A lower bound for real volume of the iterates of a divisor.
Definition 3.9. LetM be a differentiable surface. A familyΓ of curves onM is said to be very ample if for all P ∈M and for all directions D ⊂TxM, there is a curveγ∈Γon whichP is a regular point and whose tangent direction atP isD. Example 3.10. LetX be a real algebraic surface andD be a very ample real divisor onX. Then the familyV(D), as a family of curves onX(R), is a very ample family in the sense of Definition 3.9.
Theorem 3.11. LetM be a compact Riemannian surface,g:M →M be a diffeo- morphism of classC1+ε (with ε >0) with positive entropy, andΓ be a very ample family of curves on M. Then for all λ <exp(htop(g)), there exist a curve γ∈Γ and a constant C >0 such that
length(gn(γ))≥Cλn (36)
for alln∈N.
In other words, we have the following inequality:
sup
γ∈Γ
lim inf
n→+∞
1
nlog (length(gn(γ)))
≥htop(g). (37) This has to be compared with a similar result due to Newhouse [27], who con- siders manifolds of arbitrary dimension and noninvertible maps, but obtains the inequality (37) with a limit superior instead of a limit inferior (assumptions on the family Γ are also lightly different). On the other hand, the lower bound (37) is optimal whenM andg areC∞, by Yomdin’s Theorem 3.5.
Corollary 3.12. Let f be a real automorphism of a real algebraic surfaceX. For all λ <exp(htop(fR))and all very ample real divisors D on X, there existsC >0 such that
mvolR(f∗nD)≥Cλn (38)
for alln∈N.
The proof of Theorem 3.11 relies on a result due to Katok [18, S.5.9 p. 698], which asserts that the entropy of surface diffeomorphisms is well approximated by horseshoes. For definition and properties of horseshoes, we refer to [18, §6.5].
Theorem 3.13 (Katok). Let M be a compact surface, and g : M → M be a diffeomorphism of class C1+ε (with ε >0) with positive entropy. For any η > 0, there exists a horseshoeΛ for some positive iterate gk of g such that
htop(g)≤ 1
khtop(g|Λk ) +η. (39)
Proof of Theorem 3.11. Fix real numbersλand η such that 1< λ <exp(htop(g)) and 0 < η ≤htop(g)−log(λ). Let Λ be a horseshoe forG = gk satisfying (39).
Let ∆ ⊃ Λ be a “rectangle” corresponding to this horseshoe, in such a way that Λ =T
j∈ZGj(∆). The set G(∆)∩∆ has q connected components ∆1,· · ·,∆q, which are “subrectangles” crossing entirely∆downward (see Figure 1). The restric- tionG|Λ is topologically conjugate to the full-shift onqsymbols, by the conjugacy map
{1,· · ·, q}Z−→Λ (ωj)j∈Z7−→ \
j∈Z
Gj(∆ωj).
In particularhtop(G|Λ) = log(q). We denote byL the distance between the upper and lower side of∆.
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111
∆1 ∆2
G(∆)
∆
L γ
G(γ)
Figure 1. An example of horseshoe, here with q= 2.
Lemma 3.14. Let γ ⊂ ∆ be an arc crossing the rectangle ∆ downward. Then length(Gn(γ))≥qnLfor alln∈N.
Proof. It is enough to remark that the arc Gn(γ) containsqn subarcs crossing∆ downward (see Figure 1 forn= 1). This can be seen by induction onn.
Now fix a point P ∈ Λ, P = T
j∈ZGj(∆ωj). Letγ ∈ Γ be a curve that goes throughP transversally to the stable varietyWs(P)(the horizontal one). For any sequence(εj)j∈N ∈ {1,· · ·, q}N, we set (see Figure 2)
Rε1,···,εn=
n
\
j=0
G−j(∆εj). (40) The sequence (Rε1,···,εn)n∈N is a decreasing sequence of nested rectangles that converge to the curve T
j∈NG−j(∆εj). If (εn)n∈N = (ω−n)n∈N this curve is the stable varietyWs(P)(intersected with∆). Sinceγis transverse to it, there exist an integern0and a subarcγ′ ⊂γsuch thatγ′crosses the rectangleRω0,···,ω−n0 down- ward. (On Figure 2, we may chooseγ′ ⊂ R22.) Hence the arcGn0(γ′)⊂Gn0(γ)
γ Ws(P)
P
R21
R1
R2
R11
R22
R12
Figure 2. The rectanglesRε1 andRε1,ε2 for the horseshoe of Fig- ure 1.
satisfies the assumptions of Lemma 3.14, and thus length(Gn0+n(γ)) ≥ qnL for alln∈N. So if we setC′= min
L
qn0,length(Gn(γ)) qn
0≤n≤n0−1
, then length(gnk(γ))≥C′qn
=C′exp(nhtop(g|Λk ))
≥C′exp(nk(htop(g)−η))
≥C′λnk.
(41)
Sincelength(gn(γ))≤ kDg−1klength(gn+1(γ)), we get Inequality (36) by Euclidean division byk, where we have setC=C′(λkDg−1k∞)−k >0.
3.5. An exact formula for concordance whenρ(XR) = 2.
Theorem 3.15. Let X be a real algebraic surface withρ(XR) = 2. Assume that there exists a real hyperbolic type automorphismf on X. Then
α(X) =htop(fR)
htop(fC). (42)
Remark 3.16. The assumptions of the theorem imply that the surfaceX is either a torus, a K3 surface, or an Enriques surface. Indeed, as seen in Section 1.4, its minimal model is either one of these three types of surfaces, or a rational surface.
But if X is not minimal or if X is rational, then the class of the canonical divi- sorKX would be nontrivial inN1(XR;R). Since this class is preserved byf∗, this map would have 1 as an eigenvalue. This is impossible, because N1(XR;R) has dimension2and the spectral radius off∗ must be >1.
Proof of Theorem 3.15. By Theorem 3.8, it is enough to prove that any nonnegative exponentα < hhtop(fR)
top(fC) belongs toA(X). This is obvious whenhtop(fR) = 0, so we suppose thatfR has positive entropy, and we fix such an exponentα.
Lemma 3.17. Let D be a very ample real divisor onX. There existsC >0such that
mvolR(f∗nD)≥CvolC(f∗nD)α (43) for alln∈Z.
Proof. Since λ(f)α = exp(αhtop(fC)) < exp(htop(fR)), there exists, by Corol- lary 3.12, a positive numberCRsuch thatmvolR(f∗nD)≥CRλ(f)nα for alln∈N.
On the other hand there is a positive numberCCsuch thatvolC(f∗nD)≤CCλ(f)n for alln∈N(cf (27)). It follows thatmvolR(f∗nD)≥C+volC(f∗nD)αfor alln∈N, where we have setC+=CR/CCα.
Applying the same argument to f−1, there exists a positive number C− such that mvolR(f∗−nD)≥C−volC(f∗−nD)α for all n ∈ N. Hence we obtain (43),
withC= min(C+, C−).
Lemma 3.18. There are finitely many real ample divisors D1,· · ·, Dr onX such that any real ample divisor D on X is equivalent to one of the formPs
k=1f∗nDjk, withn∈Zandjk∈ {1,· · ·, r}.
Proof. On the 2-dimensional space N1(XR;R), the isometry f∗ has exactly two eigenlines D+ and D−, respectively associated with eigenvalues λ(f) and λ(f)−1. These lines are necessarily the isotropic directions of the intersection form. We choose eigenvectorsθ+ ∈ D+ andθ− ∈ D− in the closure ofPos(XR), so that this cone is bordered by half-linesR+θ+andR+θ−. Since it is preserved byf∗, the cone Amp(XR)coincides withPos(XR). The integer points in this cone correspond to classes of real ample divisors. Letθ1be such a point that we choose to be primitive, and letθ2=f∗θ1(observe thatθ2is also primitive). Denote byDthe closed convex cone ofN1(XR;R)bordered by half-linesR+θ1andR+θ2. By constructionD\{0} is a fundamental domain for the action off∗ onAmp(XR)(see Figure 3).
0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000
1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111
f∗
f∗n D Amp(XR)
R+θ− R+θ+
θ1 θ2
Figure 3. The fundamental domainD\{0}.
Denote byθ3, θ4,· · ·, θrthe entire points inside the parallelogram whose vertices are0,θ1,θ1+θ2andθ2. Any point inDcan be expressed uniquely ask1θ1+k2θ2+θj
ork1θ1+k2θ2, with(k1, k2)∈N2andj∈ {3,· · ·, r}. For all real ample divisorsD, there is n∈ Z such that f∗−n[D]∈ D, so we are done by settingD1,· · ·, Dr real ample divisors whose classes areθ1,· · ·, θr. We go back to the proof of Theorem 3.15. Let q be a positive integer such that the divisors D′1 = qD1,· · ·, Dr′ =qDr are all very ample. By Lemma 3.17, there exists a positive number C such that, for all j ∈ {1,· · · , r} and n ∈ Z, we have mvolR(f∗nD′j) ≥ CvolC(f∗nD′j)α. Let D be a real ample divisor whose Chern class is q-divisible. There are n ∈ Z and j1,· · · , js ∈ {1,· · ·, r} such
that[D] =Ps
k=1f∗n[D′jk]. Then mvolR(D)≥X
k
mvolR(f∗nD′jk) (44)
≥CX
k
volC(f∗nD′jk)α (45)
≥C X
k
volC(f∗nD′jk)
!α
(46)
=CvolC(D)α. (47)
From (45) to (46), we have used the following special case of Minkowski inequality:
s
X
k=1
|xk|
!α
≤
s
X
k=1
|xk|α ∀α∈(0,1]. (48) Hence we see thatαbelongs toA(X), and Theorem 3.15 is proved.
Remark 3.19. We do not know if concordance is achieved in Theorem 3.15.
4. Abelian Surfaces
4.1. Preliminaries. A real abelian variety X is a real algebraic variety whose underlying complex manifold X(C)is a complex torusCg/Λ. We sayreal elliptic curve when g = 1, and real abelian surface when g = 2. As we still assume thatX(R)6=∅, we are brought to the case where the antiholomorphic involutionσX
comes from the complex conjugation onCg, and the latticeΛ has the form
Λ =Zg⊕τZg, (49)
where τ ∈ Mg(C) is such that Im(τ) ∈ GLg(R) and 2Re(τ) =
Ir 0 0 0
, the integerrbeing characterized by the fact thatX(R)has2g−rconnected components (cf [31, §IV]).
A (real) homomorphism between two real abelian varieties is a holomorphic map f : X = Cg/Λ → X′ = Cg′/Λ′ which is compatible with the real struc- tures (that is, σX′ ◦f = f ◦σX) and which respects the abelian group struc- tures (this is equivalent to f(0) = 0). Such a map lifts to a unique C-linear mapF :Cg→Cg′ such thatF(Λ)⊂Λ′, whose matrix has integer coefficients (for F(Zg)⊂Λ′∩Rg′ =Zg′). We also talk about endomorphisms, isomorphisms and automorphisms of real abelian varieties. Observe that in this context, automor- phisms are asked topreserve the origin.
A (real)isogenybetween two real abelian varieties of same dimension is a homo- morphism of real abelian varieties that is surjective, which means that its matrix has maximal rank. Two real abelian varieties are said to be isogenouswhen there exists an isogeny from one to the other (this is an equivalence relation; cf [6, 1.2.6]).
Remark 4.1. The real Picard number does not change by isogeny. Indeed, any isogenyf : X → X′ gives rise to a homomorphismf∗: N1(XR′ ;Z)→N1(XR;Z) which is injective, henceρ(XR′ )≤ρ(XR); the other inequality follows by the sym- metry of the isogeny relation.
Lemma 4.2. Any real abelian surface X is isogenous to C2/Λ, where Λ has the form
Λ =Z2⊕iSZ2, (50)
the matrix S=
y1 y3
y3 y2
being symmetric positive definite. We then have
ρ(XR) = 4−dimQ(Qy1+Qy2+Qy3). (51) Proof. The existence of a real polarization onX =Cg/Λ (see [31, §IV.3]) implies that the latticeΛcan be set on the formDZ2⊕τZ2, the matrixDbeing diagonal with integer coefficients, and the matrixτ being symmetric, with S=Im(τ)posi- tive definite and2Re(τ)an integer matrix. Hence the dilation by2inC2gives rise to a real isogeny fromC2/Λ toC2/(Z2⊕iSZ2). Equality (51) comes from [6, §1,
3.4] and [31, §IV (3.4)].
Remark 4.3. As a consequence of (51), we see that the real Picard numberρ(XR) is 1, 2 or 3. In contrast, the complex Picard number ρ(XC) can also achieve the extra value4, whenX is isogenous to the square of an elliptic curve with complex multiplication (cf [6, §2 7.1]).
Now observe the following fact, which is very specific to tori.
Proposition 4.4. Let f be an automorphism of a real abelian surfaceX. Then htop(fC) = 2 htop(fR). (52) Accordingly,α(X)≤1/2 as soon asX admits real hyperbolic type automorphisms.
Proof. We lift the automorphismf to aC-linear mapF :C2→C2 whose matrix is inSL2(Z)(replacingf byf2 if necessary). IfF has spectral radius1, then it is obvious thathtop(fR) = htop(fC) = 0. Otherwise,F has two distinct eigenvaluesλ andλ−1, with|λ|>1. As aR-linear map ofC2,F has eigenvalues(λ, λ, λ−1, λ−1) (with multiplicities), thus htop(fC) = 2 log|λ| (see, for instance, [7, 2.6.4]). Re- stricted toR2,F has eigenvalues(λ, λ−1), hencehtop(fR) = log(|λ|).
The last part is a consequence of Theorem 3.8.
The aim of what follows is to prove the following theorem, which describes ex- haustively the concordance for real abelian surfaces.
Theorem 4.5. LetX be a real abelian surface. We have the following alternative:
(1) ρ(XR) = 1andα(X) = 1;
(2) ρ(XR) = 2and
(i) if the intersection form represents0 onN1(XR;Z), thenα(X) = 1, (ii) otherwise,α(X) = 1/2;
(3) ρ(XR) = 3andα(X) = 1/2.
The concordance is achieved in all cases. It equals1/2 if and only ifX admits real hyperbolic type automorphisms.
We already dealt with the caseρ(XR) = 1(cf Corollary 2.8), so we focus on the last two cases.
4.2. Invariance of concordance under isogeny.
Proposition 4.6. Let X and X′ be two isogenous real abelian varieties. Then A(X) =A(X′), and consequentlyα(X) =α(X′).
Proof. Since the isogeny relation is symmetric, it is enough to show the inclu- sion A(X) ⊂ A(X′). Let f : X′ →X be an isogeny. For K = R or C, denote byfK the induced map fromX′(K)toX(K). We take an arbitrary Kähler metric onX, and then we take its pullback onX′, so thatf is locally an isometry for the respective metrics.
Fix any α∈ A(X). There exist C > 0 and q ∈ N∗ such that any real ample divisorDonX, whose Chern class isq-divisible, satisfiesmvolR(D)≥CvolC(D)α. Since f∗ : N1(XR;Z)→ N1(XR′ ;Z)is an injective homomorphism, its image has finite index n. Let D′ be a real ample divisor on X′ whose class is nq-divisible.
Then there is a real ample divisorDonX with[D′] = [f∗D], and furthermore[D]
isq-divisible.
Any point on the curveD(R)has exactlydeg(fR)preimages, hencevolR(f∗D) = deg(fR) volR(D) by the choice of the metrics. Since f∗ realizes a bijective map between V(D) and V(D′), we deduce, taking the upper bound on V(D), that mvolR(D′) = deg(fR) mvolR(D).
By the same argument, we also have volC(D′) = deg(fC) volC(D). So if we set C′ =Cdeg(fR)/deg(fC)α, we obtain mvolR(D′)≥C′volC(D′)α. This shows
that the exponentαis contained inA(X′).
4.3. Picard number 2.
4.3.1. Hyperbolic rank 2 lattices. By definition, alattice is a free abelian group L of finite rank, equipped with a nondegenerate symmetric bilinear form ϕ taking integral values. We say the lattice ishyperbolic when the signature of the induced quadratic form onL⊗Ris(1,rank(L)−1). The determinant of the matrix ofϕin a base ofLis the same for all bases. Its absolute value is a positive integer, called thediscriminant of the lattice.
Let(L, ϕ)be a rank 2 hyperbolic lattice. ThenL⊗Rhas exactly two isotropic lines. The discriminant δ is a perfect square if and only if the quadratic form associated to ϕ represents 0, which means that there exists a nonzero isotropic point inL, or to say it otherwise both isotropic lines inL⊗Rare rational.
Suppose that δ is no perfect square. The study of Pell–Fermat equation then implies the existence of a hyperbolic isometry of L, that is, an isometry whose spectral radius is greater than 1. Such an isometry spans a finite index subgroup of the isometries ofL. To be more precise, the groupSO(L, ϕ)of direct isometries ofL(those with determinant1) is an abelian group isomorphic toZ×Z/2Z, and any infinite order element inSO(L, ϕ)is hyperbolic.
Conversely if δ is a perfect square, there is no hyperbolic isometry, and the isometry group is finite. More precisely,SO(L, ϕ) ={id,−id} ≃Z/2Z.
Example 4.7. Let X be a real algebraic surface with ρ(XR) = 2. Then the groupN1(XR;Z), equipped with the intersection form, is a rank 2 hyperbolic lat- tice.
4.3.2. Surfaces with real elliptic fibrations. LetX andY be two complex algebraic varieties. An elliptic fibration on X is a holomorphic map π : X → Y that is proper and surjective, and such that the generic fiber is an elliptic curve. When the
varietiesX,Y and the morphismπare defined overR, we call the elliptic fibration real.
Proposition 4.8. Let X be a real abelian surface with ρ(XR) = 2. The following are equivalent:
(1) the intersection form onN1(XR;Z) represents0;
(2) there exists a real elliptic fibration on X;
(3) X is isogenous to the product of two elliptic curves E1 andE2. In this case, the concordance of X equals1, and it is achieved.
Remark 4.9. The elliptic curvesE1andE2 cannot be isogenous, for otherwise the Picard number would be3.
Proof. (1)⇒(2): Let θa nonzero primitive point in N1(XR;Z) such thatθ2= 0.
After changingθ into −θ if necessary, there exists a real effective and irreducible divisor D whose class is θ (here we use the fact that Nef(XR) is the closure of Pos(XR); cf [20, 1.5.17]). By the genus formula, the arithmetic genus of D is 1. Since an abelian surface does not have any rational curve, D must be a real elliptic curve. We may suppose thatD goes through0(if not, we translate it and obtain an equivalent divisor), and thus it is a real subtorus. Now the canonical projectionπ:X→X/D is a real elliptic fibration.
(2)⇒(3)follows from the Poincaré reducibility theorem (see [12, §VI 8.1]).
(3)⇒ (1): Letf :X →E1×E2 be an isogeny. The effective divisor D given byf∗(E1× {0})has self-intersection 0, so the intersection form represents0.
As the nef cone ofE1×E2is spanned by[E1× {0}]and[{0} ×E2], the interval A(E1×E2)is equal to[0,1], by Proposition 2.7. By invariance under isogeny, we
also haveA(X) = [0,1].
4.3.3. Surfaces with no real elliptic fibration.
Theorem 4.10. Let X be a real abelian surface withρ(XR) = 2. Assume that the intersection form on N1(XR;Z)does not represent0. Then
(1) there exists a real hyperbolic type automorphism onX; (2) the concordance ofX equals1/2and it is achieved.
We use the following result (see, for instance, [2]):
Theorem 4.11 (Torelli theorem for tori). Let X be a real abelian surface and let φ be a Hodge isometry of H2(X(C);Z) that preserves the ample cone and has determinant +1. Then there exists a complex automorphism f of X(C), unique up to a sign, such that f∗ =φ. If moreover φ commutes with the involution σ∗X, thenf or f2 is a realautomorphism.
Remark 4.12.To prove the last part, it is enough to remark thatσX◦f◦σX =±f−1 by the uniqueness part.
Lemma 4.13. Let L be an free abelian group of finite rank, L′ be a finite index subgroup of L and φ′ be an automorphism of L′. Then some positive iterate φ′k extends to an automorphismφ onL.
Proof. Denote byqthe exponent of the groupL/L′, so thatqL⊂L′. Asφ′projects to an automorphism ofL′/qL′that has finite orderk, it follows thatφ′k(qL)⊂qL.
Let µq : L → qL be the isomorphism defined by θ 7→ qθ. Then the automor- phismφ=µ−1q ◦φ′k|qL◦µq satisfies the desired property.
